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NDA 2011
Exam
NDA
Year
2011
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2011 · Mathematics · Indefinite Integration

What is ∫ e^x( x + 1 2 x ) d x equal to? Where c is a constant of...

What is ∫ex(x+12x)dx\int e^x\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x equal to? Where cc is a constant of integration.

Options

  1. Axex+cx e^x+c
  2. Bex(x)+ce^x(\sqrt{x})+c
  3. C2ex(x)+c2 e^x(\sqrt{x})+c
  4. D2xex+c2 x e^x+c

Correct answer

B. ex(x)+ce^x(\sqrt{x})+c

Step-by-step solution

∫ex(x+12x)dx=∫exxdx+∫ex2xdx=x∫exdx−∫(d/dxx(x)∫exdx)dx+∫ex2xdx=exx−∫12x×exdx+∫ex2xdx=exx+c\begin{aligned} & \int e^x\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x=\int e^x \sqrt{x} d x+\int \frac{e^x}{2 \sqrt{x}} d x \\ & =\sqrt{x} \int e^x d x-\int\left(d / d x x(\sqrt{x}) \int e^x d x\right) d x+\int \frac{e^x}{2 \sqrt{x}} d x \\ & =e^x \sqrt{x}-\int \frac{1}{2 \sqrt{x}} \times e^x d x+\int \frac{e^x}{2 \sqrt{x}} d x \\ & =e^x \sqrt{x}+c\end{aligned}
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